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December  2020, 25(12): 4779-4799. doi: 10.3934/dcdsb.2020126

Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

Rajeshwari Majumdar 1,†, , Phanuel Mariano 2,‡,, , Hugo Panzo 3,**, , Lowen Peng 4,†, and  Anthony Sisti 5,†,

1. 

Department of Politics, New York University, New York, NY 10012, USA
2. 

Department of Mathematics, Union College, Schenectady, NY 12308, USA
3. 

Faculties of Electrical Engineering and Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
4. 

Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
5. 

Department of Biostatistics, Brown University, Providence, RI 02912, USA

* Corresponding author: Phanuel Mariano

† Research was supported in part by NSF Grant DMS-1262929.
‡ Research was supported in part by NSF Grant DMS-1405169, 1712427.
** Research was supported in part by the UConn Mathematics Department and a Zuckerman fellowship.

Received  April 2019 Revised  December 2019 Published  December 2020 Early access  April 2020

We consider three matrix models of order 2 with one random entry $ \epsilon $ and the other three entries being deterministic. In the first model, we let $ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ \epsilon\sim \rm{Bernoulli}\left(p\right) $ and $ p\in [0, 1] $ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy random variable and $\xi$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

Keywords: Products of random matrices, Lyapunov exponents, continued fractions, Fibonacci sequences.
Mathematics Subject Classification: Primary: 37H15; Secondary: 37M25, 60B20, 60B15, 11B39.
Citation: Rajeshwari Majumdar, Phanuel Mariano, Hugo Panzo, Lowen Peng, Anthony Sisti. Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4779-4799. doi: 10.3934/dcdsb.2020126
References:
[1]

G. Akemann, Z. Burda and M. Kieburg, Universal distribution of Lyapunov exponents for products of Ginibre matrices, J. Phys. A, 47 (2014), 395202, 35 pp. doi: 10.1088/1751-8113/47/39/395202.  Google Scholar

[2]

G. Akemann, M. Kieburg and L. Wei, Singular value correlation functions for products of Wishart random matrices, J. Phys. A, 46 (2013), 275205, 22 pp. doi: 10.1088/1751-8113/46/27/275205.  Google Scholar

[3]

Y. Benoist and J.-F. Quint, Central limit theorem for linear groups, Ann. Probab., 44 (2016), 1308-1340.  doi: 10.1214/15-AOP1002.  Google Scholar

[4]

P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Progress in Probability and Statistics, 8. Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4684-9172-2.  Google Scholar

[5]

P. Chassaing, G. Letac and M. Mora, Brocot sequences and random walks in SL(2, R), Probability Measures on Groups, VII (Oberwolfach, 1983), Lecture Notes in Math., Springer, Berlin, 1064 (1984), 36–48. doi: 10.1007/BFb0073632.  Google Scholar

[6]

J. E. Cohen and C. M. Newman, The stability of large random matrices and their products, Ann. Probab., 12 (1984), 283–310, http://links.jstor.org.ezproxy.lib.uconn.edu/sici?sici=0091-1798(198405)12:2<283:TSOLRM>2.0.CO;2-O&origin=MSN. doi: 10.1214/aop/1176993291.  Google Scholar

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P. J. Forrester, Lyapunov exponents for products of complex Gaussian random matrices, J. Stat. Phys., 151 (2013), 796-808.  doi: 10.1007/s10955-013-0735-7.  Google Scholar

[8]

P. J. Forrester, Asymptotics of finite system Lyapunov exponents for some random matrix ensembles, J. Phys. A, 48 (2015), 215205, 17 pp. doi: 10.1088/1751-8113/48/21/215205.  Google Scholar

[9]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.  doi: 10.1214/aoms/1177705909.  Google Scholar

[10]

H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32.  doi: 10.1007/BF02760620.  Google Scholar

[11]

A. Goswami, Random continued fractions: A Markov chain approach, Econom. Theory, 23 (2004), 85–105, https://doi.org/10.1007/BF02760620. doi: 10.1007/s00199-002-0333-4.  Google Scholar

[12]

É. Janvresse, B. Rittaud and T. de la Rue, How do random Fibonacci sequences grow?, Probab. Theory Related Fields, 142 (2008), 619-648.  doi: 10.1007/s00440-007-0117-7.  Google Scholar

[13]

É. Janvresse, B. Rittaud and T. de la Rue, Growth rate for the expected value of a generalized random Fibonacci sequence, J. Phys. A, 42 (2009), 085005, 18 pp. doi: 10.1088/1751-8113/42/8/085005.  Google Scholar

[14]

É. Janvresse, B. Rittaud and T. de la Rue, Almost-sure growth rate of generalized random Fibonacci sequences, Ann. Inst. Henri Poincaré Probab. Stat., 46 (2010), 135-158.  doi: 10.1214/09-AIHP312.  Google Scholar

[15]

V. Kargin, On the largest Lyapunov exponent for products of Gaussian matrices, J. Stat. Phys., 157 (2014), 70-83.  doi: 10.1007/s10955-014-1077-9.  Google Scholar

[16]

M. Kieburg and H. Kösters, Products of random matrices from polynomial ensembles, Ann. Inst. Henri Poincaré Probab. Stat., 55 (2019), 98-126.  doi: 10.1214/17-AIHP877.  Google Scholar

[17]

R. Lima and M. Rahibe, Exact Lyapunov exponent for infinite products of random matrices, J. Phys. A, 27 (1994), 3427-3437.  doi: 10.1088/0305-4470/27/10/019.  Google Scholar

[18]

J. Marklof, Y. Tourigny and L. Wolowski, Explicit invariant measures for products of random matrices, Trans. Amer. Math. Soc., 360 (2008), 3391-3427.  doi: 10.1090/S0002-9947-08-04316-X.  Google Scholar

[19]

C. M. Newman, The distribution of Lyapunov exponents: Exact results for random matrices, Comm. Math. Phys., 103 (1986), 121–126, http://projecteuclid.org/euclid.cmp/1104114627. doi: 10.1007/BF01464284.  Google Scholar

[20]

Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., Springer, Berlin, 1486 (1991), 64– 80. doi: 10.1007/BFb0086658.  Google Scholar

[21]

Y. Peres, Domains of analytic continuation for the top Lyapunov exponent, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 131–148, http://www.numdam.org/item?id=AIHPB_1992__28_1_131_0.  Google Scholar

[22]

M. Pollicott, Maximal Lyapunov exponents for random matrix products, Invent. Math., 181 (2010), 209-226.  doi: 10.1007/s00222-010-0246-y.  Google Scholar

[23]

V. Yu. Protasov and R. M. Jungers, Lower and upper bounds for the largest Lyapunov exponent of matrices, Linear Algebra Appl., 438 (2013), 4448-4468.  doi: 10.1016/j.laa.2013.01.027.  Google Scholar

[24]

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Adv. in Math., 32 (1979), 68-80.  doi: 10.1016/0001-8708(79)90029-X.  Google Scholar

[25]

D. Viswanath, Random Fibonacci sequences and the number $1.13198824\ldots$, Math. Comp., 69 (2000), 1131-1155.  doi: 10.1090/S0025-5718-99-01145-X.  Google Scholar

show all references

References:
[1]

G. Akemann, Z. Burda and M. Kieburg, Universal distribution of Lyapunov exponents for products of Ginibre matrices, J. Phys. A, 47 (2014), 395202, 35 pp. doi: 10.1088/1751-8113/47/39/395202.  Google Scholar

[2]

G. Akemann, M. Kieburg and L. Wei, Singular value correlation functions for products of Wishart random matrices, J. Phys. A, 46 (2013), 275205, 22 pp. doi: 10.1088/1751-8113/46/27/275205.  Google Scholar

[3]

Y. Benoist and J.-F. Quint, Central limit theorem for linear groups, Ann. Probab., 44 (2016), 1308-1340.  doi: 10.1214/15-AOP1002.  Google Scholar

[4]

P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Progress in Probability and Statistics, 8. Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4684-9172-2.  Google Scholar

[5]

P. Chassaing, G. Letac and M. Mora, Brocot sequences and random walks in SL(2, R), Probability Measures on Groups, VII (Oberwolfach, 1983), Lecture Notes in Math., Springer, Berlin, 1064 (1984), 36–48. doi: 10.1007/BFb0073632.  Google Scholar

[6]

J. E. Cohen and C. M. Newman, The stability of large random matrices and their products, Ann. Probab., 12 (1984), 283–310, http://links.jstor.org.ezproxy.lib.uconn.edu/sici?sici=0091-1798(198405)12:2<283:TSOLRM>2.0.CO;2-O&origin=MSN. doi: 10.1214/aop/1176993291.  Google Scholar

[7]

P. J. Forrester, Lyapunov exponents for products of complex Gaussian random matrices, J. Stat. Phys., 151 (2013), 796-808.  doi: 10.1007/s10955-013-0735-7.  Google Scholar

[8]

P. J. Forrester, Asymptotics of finite system Lyapunov exponents for some random matrix ensembles, J. Phys. A, 48 (2015), 215205, 17 pp. doi: 10.1088/1751-8113/48/21/215205.  Google Scholar

[9]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.  doi: 10.1214/aoms/1177705909.  Google Scholar

[10]

H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32.  doi: 10.1007/BF02760620.  Google Scholar

[11]

A. Goswami, Random continued fractions: A Markov chain approach, Econom. Theory, 23 (2004), 85–105, https://doi.org/10.1007/BF02760620. doi: 10.1007/s00199-002-0333-4.  Google Scholar

[12]

É. Janvresse, B. Rittaud and T. de la Rue, How do random Fibonacci sequences grow?, Probab. Theory Related Fields, 142 (2008), 619-648.  doi: 10.1007/s00440-007-0117-7.  Google Scholar

[13]

É. Janvresse, B. Rittaud and T. de la Rue, Growth rate for the expected value of a generalized random Fibonacci sequence, J. Phys. A, 42 (2009), 085005, 18 pp. doi: 10.1088/1751-8113/42/8/085005.  Google Scholar

[14]

É. Janvresse, B. Rittaud and T. de la Rue, Almost-sure growth rate of generalized random Fibonacci sequences, Ann. Inst. Henri Poincaré Probab. Stat., 46 (2010), 135-158.  doi: 10.1214/09-AIHP312.  Google Scholar

[15]

V. Kargin, On the largest Lyapunov exponent for products of Gaussian matrices, J. Stat. Phys., 157 (2014), 70-83.  doi: 10.1007/s10955-014-1077-9.  Google Scholar

[16]

M. Kieburg and H. Kösters, Products of random matrices from polynomial ensembles, Ann. Inst. Henri Poincaré Probab. Stat., 55 (2019), 98-126.  doi: 10.1214/17-AIHP877.  Google Scholar

[17]

R. Lima and M. Rahibe, Exact Lyapunov exponent for infinite products of random matrices, J. Phys. A, 27 (1994), 3427-3437.  doi: 10.1088/0305-4470/27/10/019.  Google Scholar

[18]

J. Marklof, Y. Tourigny and L. Wolowski, Explicit invariant measures for products of random matrices, Trans. Amer. Math. Soc., 360 (2008), 3391-3427.  doi: 10.1090/S0002-9947-08-04316-X.  Google Scholar

[19]

C. M. Newman, The distribution of Lyapunov exponents: Exact results for random matrices, Comm. Math. Phys., 103 (1986), 121–126, http://projecteuclid.org/euclid.cmp/1104114627. doi: 10.1007/BF01464284.  Google Scholar

[20]

Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., Springer, Berlin, 1486 (1991), 64– 80. doi: 10.1007/BFb0086658.  Google Scholar

[21]

Y. Peres, Domains of analytic continuation for the top Lyapunov exponent, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 131–148, http://www.numdam.org/item?id=AIHPB_1992__28_1_131_0.  Google Scholar

[22]

M. Pollicott, Maximal Lyapunov exponents for random matrix products, Invent. Math., 181 (2010), 209-226.  doi: 10.1007/s00222-010-0246-y.  Google Scholar

[23]

V. Yu. Protasov and R. M. Jungers, Lower and upper bounds for the largest Lyapunov exponent of matrices, Linear Algebra Appl., 438 (2013), 4448-4468.  doi: 10.1016/j.laa.2013.01.027.  Google Scholar

[24]

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Adv. in Math., 32 (1979), 68-80.  doi: 10.1016/0001-8708(79)90029-X.  Google Scholar

[25]

D. Viswanath, Random Fibonacci sequences and the number $1.13198824\ldots$, Math. Comp., 69 (2000), 1131-1155.  doi: 10.1090/S0025-5718-99-01145-X.  Google Scholar

Figure 1.  Histogram
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Figure 2.  CDF
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Figure 3.  $ n = 1\, 000\, 000 $
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Figure 4.  $ \lambda(\xi) $ vs. $ \xi $
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Figure 5.  $ k = 0.01 $, $ n = 1000 $, $ m = 1\, 000\, 000 $
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Figure 6.  $ k = 0.25 $, $ n = 1000 $, $ m = 5\, 000\, 000 $
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Figure 7.  $ k = 0.01 $, $ n = 1000 $, $ m = 1\, 000\, 000 $
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